Convolution Examples

Introduction

The best way to master convolution is through worked examples. This section presents several complete solutions using both analytical and graphical methods. Each example demonstrates a different signal type or technique, building your ability to handle any convolution problem you encounter in examinations or practice. Pay attention to how the limits of integration are determined — this is where most errors occur.

Example 1: Two Causal Exponentials

Compute $y(t) = e^{-t}u(t) * e^{-2t}u(t)$

Solution: Using the convolution integral: $$y(t) = \int_{-\infty}^{\infty} e^{-\tau}u(\tau) \cdot e^{-2(t-\tau)}u(t-\tau) d\tau$$

For $t > 0$: Both unit steps are 1 when $0 \leq \tau \leq t$: $$y(t) = \int_0^t e^{-\tau} \cdot e^{-2(t-\tau)} d\tau = e^{-2t}\int_0^t e^{\tau} d\tau = e^{-2t}[e^t - 1] = e^{-t} - e^{-2t}$$

For $t < 0$: No overlap, $y(t) = 0$.

Final answer: $y(t) = (e^{-t} - e^{-2t})u(t)$

Verification: At $t = 0$: $y(0) = 1 - 1 = 0$ ✓ (both signals start at $t=0$, so output should start at 0). As $t \to \infty$: $y \to 0$ ✓ (stable system). Area check: $(1)(1/2) = 1/2$; $\int_0^\infty(e^{-t}-e^{-2t})dt = 1 - 1/2 = 1/2$ ✓.

Example 2: Rectangular Pulse with Exponential

Compute $y(t) = [u(t) - u(t-1)] * e^{-t}u(t)$

The input is a unit rectangle from $t = 0$ to $t = 1$.

$$y(t) = \int_0^{\min(t,1)} e^{-(t-\tau)} d\tau$$

Region 1 ($0 \leq t < 1$): Upper limit is $t$: $$y(t) = \int_0^t e^{-(t-\tau)}d\tau = 1 - e^{-t}$$

Region 2 ($t \geq 1$): Upper limit is 1: $$y(t) = \int_0^1 e^{-(t-\tau)}d\tau = e^{-t}\int_0^1 e^{\tau}d\tau = e^{-t}(e-1) = (e-1)e^{-t}$$

Final answer: $y(t) = (1-e^{-t})u(t) - (1-e^{-(t-1)})u(t-1)$ or equivalently the piecewise expression above.

Example 3: Discrete-Time — Finite Sequences

Compute $y[n] = x[n] * h[n]$ where $x[n] = \{1, 2, 1\}$ for $n = 0,1,2$ and $h[n] = \{1, -1\}$ for $n = 0,1$.

Tabular method:

• $x[0] \times h$: contributes $\{1, -1\}$ starting at $n = 0$
• $x[1] \times h$: contributes $\{2, -2\}$ starting at $n = 1$
• $x[2] \times h$: contributes $\{1, -1\}$ starting at $n = 2$

Sum column by column:

• $y[0] = 1$
• $y[1] = -1 + 2 = 1$
• $y[2] = -2 + 1 = -1$
• $y[3] = -1$

Result: $y[n] = \{1, 1, -1, -1\}$ for $n = 0, 1, 2, 3$. Length = $3 + 2 - 1 = 4$ ✓.

Example 4: Signal with Itself (Autocorrelation Related)

Compute $y(t) = [u(t) - u(t-T)] * [u(t) - u(t-T)]$

Two identical rectangular pulses of width $T$ and height 1.

Using the known result for convolution of two rectangles: the result is a triangle.

$$y(t) = \begin{cases} t, & 0 \leq t < T \\ 2T - t, & T \leq t < 2T \\ 0, & \text{otherwise} \end{cases}$$

Peak value: $T$ (at $t = T$). Duration: $2T$ (double the input duration). The convolution of any signal with itself always produces a symmetric (even) function centered at the origin if the signal is symmetric, which relates to the autocorrelation.

Example 5: Discrete Exponential with Unit Step

Compute $y[n] = (0.5)^n u[n] * u[n]$

$$y[n] = \sum_{k=0}^{n} (0.5)^k \cdot 1 = \sum_{k=0}^{n} (0.5)^k = \frac{1-(0.5)^{n+1}}{1-0.5} = 2(1-(0.5)^{n+1})$$

For $n \geq 0$: $y[n] = 2 - (0.5)^n$

As $n \to \infty$: $y[n] \to 2$ (the step response of the first-order system approaches the DC gain $H(e^{j0}) = \frac{1}{1-0.5} = 2$) ✓.

Example 6: Convolution with an Impulse Train

Compute $x(t) * \sum_{k=0}^{2} \delta(t - kT)$

Using the shifting property of convolution with impulses: $$y(t) = x(t) + x(t-T) + x(t-2T)$$

The result is three copies of $x(t)$ at intervals of $T$. This is the principle behind FIR filter implementation: the output is a weighted sum of delayed input copies.

Verification Techniques

Always verify your convolution results using these checks:

1. Length check: Output duration = sum of input durations
2. Area check: $\int y(t)dt = (\int x dt) \times (\int h dt)$
3. Initial value: $y(t_{start})$ should be computable from the leading edges
4. Final value: For stable systems with step input, $y(\infty) = H(j0) \cdot x_{DC}$
5. Continuity: Convolution of bounded signals produces a continuous result

Key Takeaways

• Choose the computation method based on signal types: analytical for exponentials, graphical for piecewise signals, tabular for finite discrete sequences
• Always identify the limits of integration carefully — this is where errors occur
• Output length = $N + M - 1$ (discrete) or $T_x + T_h$ (continuous)
• Verify results with area checks, initial/final values, and continuity
• Convolution of two rectangles gives a triangle; two exponentials give a difference of exponentials